As we have mentioned before, the problem of quantum channel
discrimination collapses to the problem of quantum state discrimination when
we fix the input state. As a result, mathematical tools such as the quantum
fidelity and the QCB are fundamental in our analysis. Despite it has been
introduced only very recently, the QCB has been already a crucial tool in
several areas of quantum information: it has been exploited as a
distinguishability measure between qubits and single-mode Gaussian states
[1]}, [2]}, to evaluate the degree of nonclassicality for
single-mode Gaussian states [3]} or the polarization of a two-mode
state [4]}. It has also been applied in the theory of quantum phase
transitions to distinguish between different phases of the \(XY\)  model at
finite temperature [5]}, and to the discrimination of two ground
states or two thermal states in the quantum Ising model [6]}. For
continuous variables systems the quantum discrimination of Gaussian states is
a central point in view of their experimental accessibility and their
relatively simple mathematical description [7]}, [8]}. In fact, in
the case of Gaussian states, the QCB can be computed from their first and the
second statistical moments. A first formula, valid for single-mode Gaussian
states, was derived in [1]}. Later, Ref. [10]} provided a
general closed formula for multimode Gaussian states, relating the QCB bound
to their symplectic spectra. Furthermore, from these spectra one can derive
larger upper bounds which are easier to compute than the QCB [10]}.
